2 Answers
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The "primary" aliens at the heart of the conspiracy arc are the greys, which include the colonists (but not all seem intent on colonizing Earth) and, apparently, the long-clawed aliens (a different stage of their life cycle).

The faceless aliens/bounty hunters may or may not be the same race as the greys; at the very least the latter certainly work closely with them (at least the ones we've seen), although the former oppose the colonists. Their shape-shifting tendencies make it hard to be sure what race they truly belong to, though.

The purity -- aka "black oil" or "black cancer" -- is apparently an intelligent virus. However, given that infection by purity can result in the host being impregnated with a long-clawed alien (which in turn sheds its skin later to become a grey), this may be just another stage of the natural life cycle of the greys, rather than a distinct alien species.

So those are the ones involved in the main conspiracy story arc throughout the series -- anywhere from 1 to 4 distinct species of extra-terrestrials.

The "monster of the week" episodes, however, have given us several others, including the Kindred, the gaseous Martians, a spider-like creature, and more -- including multiple MotW episodes where the nature of the creature in question is never quite answered, and it may be an alien -- or it may not be (for example, the ice worm parasite, which Mulder theorized to have come to Earth on a meteorite, but that was never confirmed nor refuted).

So, yes, Earth is being visited by multiple races from at least (and most likely many more than) 2 different planets.

Greys (aka Reticulians), supposedly from the planet Reticulum. The Colonists are a group of Greys, but not all Greys are Colonists.

The faceless alien rebels and alien bounty hunters are both capable of shapeshifting. It is not 100% certain whether they're the same species, nor is it certain whether they're actually Greys. The faceless aliens are explicitly not infected with Purity, and have mutilated themselves to prevent infection from occurring.

Purity (aka black oil, aka black cancer) is a sentient alien virus. Purity has infected every (or nearly every) species in the galaxy. The black oil is capable of controlling the host body, and it's used by the Colonists to reproduce.

The Kindred are a group of aliens capable of changing gender at-will, who disguised themselves as a sort of Amish-esque community. They fled Earth when Mulder & Scully got too close, and only appeared in one episode, so there's not much more info about them. (It's possible that the Kindred are the same species as the faceless aliens and/or bounty hunters, who are capable of changing genders if they need to.)

Martians are gaseous life forms from Mars (big surprise). They are parasitic (although they can survive without a host), and can travel through space without a vehicle.

The mamalian corpse in the pilot episode is of presumably alien origin, and does not match the other alien species.

The spider in Travelers and parasitic ice worm in Ice are non-humanoids of presumably alien origin.